Integrand size = 23, antiderivative size = 298 \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b n \sqrt {d+e x}}{4 x^2}-\frac {3 b e n \sqrt {d+e x}}{8 d x}-\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{8 d^{3/2}}-\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{4 d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}+\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{2 d^{3/2}}+\frac {b e^2 n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{4 d^{3/2}} \]
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Time = 0.23 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {43, 44, 65, 214, 2392, 12, 14, 6131, 6055, 2449, 2352} \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{4 d^{3/2}}-\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{8 d^{3/2}}+\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{2 d^{3/2}}+\frac {b e^2 n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{4 d^{3/2}}-\frac {b n \sqrt {d+e x}}{4 x^2}-\frac {3 b e n \sqrt {d+e x}}{8 d x} \]
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Rule 12
Rule 14
Rule 43
Rule 44
Rule 65
Rule 214
Rule 2352
Rule 2392
Rule 2449
Rule 6055
Rule 6131
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}-(b n) \int \frac {-\sqrt {d} \sqrt {d+e x} (2 d+e x)+e^2 x^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 d^{3/2} x^3} \, dx \\ & = -\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}-\frac {(b n) \int \frac {-\sqrt {d} \sqrt {d+e x} (2 d+e x)+e^2 x^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x^3} \, dx}{4 d^{3/2}} \\ & = -\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}-\frac {(b n) \int \left (-\frac {2 d^{3/2} \sqrt {d+e x}}{x^3}-\frac {\sqrt {d} e \sqrt {d+e x}}{x^2}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x}\right ) \, dx}{4 d^{3/2}} \\ & = -\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}+\frac {1}{2} (b n) \int \frac {\sqrt {d+e x}}{x^3} \, dx+\frac {(b e n) \int \frac {\sqrt {d+e x}}{x^2} \, dx}{4 d}-\frac {\left (b e^2 n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2}} \\ & = -\frac {b n \sqrt {d+e x}}{4 x^2}-\frac {b e n \sqrt {d+e x}}{4 d x}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}+\frac {1}{8} (b e n) \int \frac {1}{x^2 \sqrt {d+e x}} \, dx-\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 d^{3/2}}+\frac {\left (b e^2 n\right ) \int \frac {1}{x \sqrt {d+e x}} \, dx}{8 d} \\ & = -\frac {b n \sqrt {d+e x}}{4 x^2}-\frac {3 b e n \sqrt {d+e x}}{8 d x}-\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{4 d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}+\frac {(b e n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 d}+\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x}\right )}{2 d^2}-\frac {\left (b e^2 n\right ) \int \frac {1}{x \sqrt {d+e x}} \, dx}{16 d} \\ & = -\frac {b n \sqrt {d+e x}}{4 x^2}-\frac {3 b e n \sqrt {d+e x}}{8 d x}-\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{4 d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}+\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{2 d^{3/2}}-\frac {(b e n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 d}-\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x}\right )}{2 d^2} \\ & = -\frac {b n \sqrt {d+e x}}{4 x^2}-\frac {3 b e n \sqrt {d+e x}}{8 d x}-\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{8 d^{3/2}}-\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{4 d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}+\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{2 d^{3/2}}+\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )}{2 d^{3/2}} \\ & = -\frac {b n \sqrt {d+e x}}{4 x^2}-\frac {3 b e n \sqrt {d+e x}}{8 d x}-\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{8 d^{3/2}}-\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{4 d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}+\frac {b e^2 n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{2 d^{3/2}}+\frac {b e^2 n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )}{4 d^{3/2}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {8 a d^{3/2} \sqrt {d+e x}+4 b d^{3/2} n \sqrt {d+e x}+4 a \sqrt {d} e x \sqrt {d+e x}+6 b \sqrt {d} e n x \sqrt {d+e x}+2 b e^2 n x^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+8 b d^{3/2} \sqrt {d+e x} \log \left (c x^n\right )+4 b \sqrt {d} e x \sqrt {d+e x} \log \left (c x^n\right )+2 a e^2 x^2 \log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 b e^2 x^2 \log \left (c x^n\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-b e^2 n x^2 \log ^2\left (\sqrt {d}-\sqrt {d+e x}\right )-2 a e^2 x^2 \log \left (\sqrt {d}+\sqrt {d+e x}\right )-2 b e^2 x^2 \log \left (c x^n\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )+b e^2 n x^2 \log ^2\left (\sqrt {d}+\sqrt {d+e x}\right )+2 b e^2 n x^2 \log \left (\sqrt {d}+\sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )-2 b e^2 n x^2 \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )-2 b e^2 n x^2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+2 b e^2 n x^2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{16 d^{3/2} x^2} \]
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\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}}{x^{3}}d x\]
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\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x}}{x^{3}}\, dx \]
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\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,\sqrt {d+e\,x}}{x^3} \,d x \]
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